Radon transforms are the theoretical foundation of many medical imaging and remote sensing application. Certainly the most well known instance is the classical Radon transform, which integrates a function over hyperplanes. Among others, inversion of the classical Radon transform is important for classical transmission computed tomography and has been studied
in many textbooks (see, for example, [1]}, [2]}). Closed form reconstruction formulas are known for a long time and have first been derived already in 1917 by J. Radon [3]}.
Another Radon transform that has been studied in detail
more recently is the spherical Radon transform.
This transform integrates a function over spherical surfaces (for some restricted centers of integration) and is, among others, important for photo- and thermoacoustic tomography [4]}. Closed form reconstruction formulas for planar and spherical center sets have been found in
[5]}, [6]}, [7]}, [8]}, [9]}. The conical Radon transform, on the other hand,
is much less studied. In particular, closed form reconstruction formulas have only been known for the case \(d= 2\) , see [10]}, [11]}.
In this paper we derived such reconstruction formulas for arbitrary dimension \(d\ge 2\) .
For computed tomography with Compton cameras [12]}, [13]}, the three dimensional case is of course the most relevant one. In this case, our reconstruction formulas have a particularly simple structure and consist of an application of the Laplacian followed by a conical back-projection. The numerical implementation seems quite straight forward following the ones of the classical or the spherical Radon transform (see, for example, [8]}, [2]}). Numerical studies, however, will be subject of future research.
